Finding Least Common Multiple (LCM)

Study Notes

Definition and Concept

The least common multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of all the integers.
It's the smallest number divisible by all the given numbers without any remainder.

Finding the LCM

Methods for finding the LCM include:
- Listing multiples: List multiples of each number until a common multiple is found. The smallest is the LCM.
- Prime factorization: Find the prime factorization of each number. The LCM is the product of the highest power of each prime factor found in any factorization.
- Using the relationship with the greatest common divisor (GCD): The product of two numbers equals the product of their LCM and GCD.
  - LCM(a, b) * GCD(a, b) = a * b

Properties of LCM

The LCM of two or more integers is always greater than or equal to the greatest of the integers.
The LCM of 1 and any integer is the integer itself.
The LCM of two relatively prime numbers (common factors only 1) is their product.
The LCM of two numbers is always divisible by both of the numbers.

Examples

Finding the LCM of 4 and 6:
- Multiples of 4: 4, 8, 12, 16, ...
- Multiples of 6: 6, 12, 18, 24, ...
- The smallest common multiple is 12. Therefore, LCM(4, 6) = 12.
Finding the LCM of 12 and 18 using prime factorization:
- Prime factorization of 12: 2² * 3
- Prime factorization of 18: 2 * 3²
- Highest power of 2: 2²
- Highest power of 3: 3²
- LCM(12, 18) = 2² * 3² = 4 * 9 = 36.
Finding the LCM of 8 and 12 using the relationship with GCD:
- GCD(8, 12) = 4
- LCM(8, 12) = (8 * 12) / GCD(8, 12) = (8 * 12) / 4 = 96 / 4 = 24

Applications of LCM

Scheduling: Determining the frequency of events occurring at regular intervals.
Fraction operations: Finding a common denominator for adding or subtracting fractions.
Number theory problems: Solving problems involving multiples of numbers.
Finding the time when two events occur simultaneously.

LCM of more than two numbers

To find the LCM of more than two numbers, use prime factorization. Identify the highest power of each prime factor in any of the factorizations.
Example: Finding LCM(6, 8, 10)
- Prime factorization of 6: 2 * 3
- Prime factorization of 8: 2³
- Prime factorization of 10: 2 * 5
- Highest power of 2: 2³
- Highest power of 3: 3¹
- Highest power of 5: 5¹
- LCM(6, 8, 10) = 2³ * 3 * 5 = 8 * 3 * 5 = 120

Distinguishing from GCD

GCD (greatest common divisor) is the largest factor common to two or more numbers.
LCM is the smallest multiple common to two or more numbers.
GCD is concerned with common factors.
LCM is concerned with common multiples.

Description

This quiz covers the definition, methods, and properties of the least common multiple (LCM) of integers. Explore various techniques such as listing multiples, prime factorization, and the relationship between LCM and GCD. Test your understanding of how to find the LCM and its significance in mathematics.